Geometric and Information-Geometric Priors for Data-Driven Decision Making

The rapid advancement of Machine Learning has transitioned from solving isolated, static tasks toward developing autonomous agents capable of operating in complex, dynamic environments. Imitation Learning (IL) and Reinforcement Learning (RL) provide the standard frameworks for such data-driven decision making, enabling agents to learn optimal behaviors through direct interaction or imitation of an expert. For these agents to
succeed in real-world applications, they must actively interact with their surroundings and observe the resulting changes to achieve specific goals.

Yet, the efficacy of current frameworks is often bottlenecked by ”black box” representations that do not exploit the spatial, topological, and distributional structures inherent in physical and optimization tasks. Standard methods tend to treat environment features and agent decisions as flat vectors or images, ignoring crucial underlying symmetries and constraints. We argue that Geometric Deep Learning (GDL) and Information Geometry (IG) provide
the necessary structure for efficient, robust, and generalizable decision-making when integrated into autonomous agents’ deployment and optimization.
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This thesis addresses three primary challenges at the intersection of geometry and agentic behavior. We first consider the challenge of capturing versatile human behavior in IL, which is often hindered by absolute coordinate systems that make demonstrated and recorded human behavior appear inconsistent. Using geometric behavioral descriptors and information-geometric projections, our proposed method captures distinct modes of expert behavior, which enables agents to robustly generalize to novel contexts in versatile ways. Next, we consider the discretization schemes of traditional mesh-based physical simulation. By treating meshes as dynamic geometric graphs, we develop both RL and IL algorithms that reason over highly irregular domains. These methods automate adaptive
meshing, focusing computational resources on critical regions for high-fidelity simulation. Finally, we move our attention to the optimization of agents in high-dimensional action spaces. We derive a trust-region method utilizing information-geometric projections that limit the update step size during training. This approach leverages the intrinsic manifold of the agent’s behavior distribution to ensure that updates remain reliable over the course
of training.

Together, these contributions demonstrate that grounding representation and optimization in geometric principles leads to more effective and stable learning. By aligning agent representations and optimization schemes with the geometry of the underlying problem, this work facilitates the development of more adaptable and structurally aware autonomous systems.